A fan is running at $3000 \mathrm{rpm}.$ It is switched off. It comes to rest by uniformly decreasing its angular speed in $10 \mathrm{seconds}.$ The total number of revolutions in this period. 

Three point particles $P\mathit{,}\mathit{ }Q\mathit{,}\mathit{ }R$ move in a circle of radius $r$ with different but constant speeds. They start moving at $t=0$ from their initial positions as shown in the figure. The angular velocities (in $\mathrm{rad} {\sec }^{-1}$) of $P\mathit{,}\mathit{ }Q$ and $R$ are $5\mathrm{\pi }, 2\mathrm{\pi }$ and $3\pi$ respectively, in the same sense. The time at which they all meet is:

A spot light $S$ rotates in a horizontal plane with a constant angular velocity of $0.1 \mathrm{rad} {\mathrm{s}}^{-1}$. The spot of light $P$ moves along the wall at a distance $3 \mathrm{m}$. What is the velocity of the spot $P$ when $\mathrm{\theta } = 45°$?

A particle $A$ moves along a circle of radius $R=50 \mathrm{cm}$ so that its radius vector $r$ relative to the fixed point $O$ (figure) rotates with the constant angular velocity $\mathrm{\omega }=0.40 \mathrm{rad} {\mathrm{s}}^{-1}$. Then modulus $v$ of the velocity of the particle, and the modulus $a$ of its total acceleration will be

A large mass $M$ is stationary at the end of a light string that passes through a smooth fixed ring to a small mass $m$ that moves around in a horizontal circular path. If $l$ is the length of the string from $m$ to the top end of the tube and $\theta$ is angle between this part and vertical part of the string as shown in the figure, then the time taken by $m$ to complete one circle is equal to,